If $X \sim \mathcal{N}(\mu, \sigma^2)$, then \begin{equation} \mathcal{I}\left(\mu, \sigma^2\right)=\left(\begin{array}{cc} \frac{1}{2\mathcal{H}(X) - \log(2 \pi e)} & 0 \\ 0 & \frac{2}{\left(2\mathcal{H}(X) - \log(2 \pi e)\right)^2} \end{array}\right) \label{eq:fisher-information-matrix-gaussian-entropy} \end{equation}
What about the multivariate case ?
The Fischer Information for $X \sim \mathcal{N}_d(\mu(\beta), \Sigma^2(\theta))$ is given by
\begin{aligned} \mathcal{I}(\beta)_{m, n} & =\frac{\partial \mu^{\top}}{\partial \beta_m} \Sigma^{-1} \frac{\partial \mu}{\partial \beta_n} \\ \mathcal{I}(\theta)_{m, n} & =\frac{1}{2} \operatorname{tr}\left(\Sigma^{-1} \frac{\partial \Sigma}{\partial \theta_m} \Sigma^{-1} \frac{\partial \Sigma}{\partial \theta_n}\right) \end{aligned}
and the entropy is given by
\begin{equation} \mathcal{H}(X)=\frac{1}{2} \ln |\Sigma^2|+\frac{d}{2}(1+\ln (2 \pi)) \end{equation}
Do we have a relation between $|\Sigma^2|$ and the norm of $\mathcal{I}$ or something ? Like $|\Sigma^2|$ going to $\infty$ implies $\mathcal{I}$ going to zero ?
More generally, I'd like to show entropy increase regularity even when $\mu$ and $\Sigma$ are functional. A simpler case (e.g. $\Sigma$ is diagonal) is also welcomed.
APPENDIX:
$\begin{array}{l} \frac{\partial \mu}{\partial \theta_m}= {\left[\begin{array}{cccc}\frac{\partial \mu_1}{\partial \theta_m} & \frac{\partial \mu_2}{\partial \theta_m} & \cdots & \frac{\partial \mu_N}{\partial \theta_m}\end{array}\right]^{\top} ; } \\ \frac{\partial \Sigma}{\partial \theta_m}=\left[\begin{array}{cccc}\frac{\partial \Sigma_{1,1}}{\partial \theta_m} & \frac{\partial \Sigma_{1,2}}{\partial \theta_m} & \cdots & \frac{\partial \Sigma_{1, N}}{\partial \theta_m} \\ \frac{\partial \Sigma_{2,1}}{\partial \theta_m} & \frac{\partial \Sigma_{2,2}}{\partial \theta_m} & \cdots & \frac{\partial \Sigma_{2, N}}{\partial \theta_m} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial \Sigma_{N, 1}}{\partial \theta_m} & \frac{\partial \Sigma_{N, 2}}{\partial \theta_m} & \cdots & \frac{\partial \Sigma_{N, N}}{\partial \theta_m}\end{array}\right] .\end{array}$